Problem: Simplify and expand the following expression: $ \dfrac{2n}{5n + 9}-\dfrac{-4}{n - 5} $
Explanation: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5n + 9)(n - 5)$ Multiply the first term by $\dfrac{n - 5}{n - 5}$ $ \begin{align*} \dfrac{2n}{5n + 9} \times \dfrac{n - 5}{n - 5} & = \dfrac{(2n)(n - 5)}{(5n + 9)(n - 5)} \\ & = \dfrac{2n^2 - 10n}{(5n + 9)(n - 5)}\end{align*} $ Multiply the second term by $\dfrac{5n + 9}{5n + 9}$ $ \begin{align*} \dfrac{-4}{n - 5} \times \dfrac{5n + 9}{5n + 9} & = \dfrac{(-4)(5n + 9)}{(n - 5)(5n + 9)} \\ & = \dfrac{-20n - 36}{(n - 5)(5n + 9)}\end{align*} $ Now we have: $ = \dfrac{2n^2 - 10n}{(5n + 9)(n - 5)} - \dfrac{-20n - 36}{(n - 5)(5n + 9)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{2n^2 - 10n - (-20n - 36)}{(5n + 9)(n - 5)} $ $ = \dfrac{2n^2 - 10n + 20n + 36}{(5n + 9)(n - 5)} $ $ = \dfrac{2n^2 + 10n + 36}{(5n + 9)(n - 5)}$ Expand the denominator: $ = \dfrac{2n^2 + 10n + 36}{5n^2 - 16n - 45}$